Question

find gh.

(image of a right triangle with right angle at h, vertex i, angle at i is 30°, length of ih is 7√3, need to find gh. write your answer in simplified, rationalized form. do not round. gh = )

Explanation:

Step1: Identify triangle type and trigonometric ratio

We have a right - triangle \( \triangle GHI \) with \( \angle H = 90^{\circ} \), \( \angle I=30^{\circ} \) and \( HI = 7\sqrt{3} \). We want to find \( GH \). We can use the tangent function, where \( \tan(\theta)=\frac{\text{opposite}}{\text{adjacent}} \). For \( \angle I = 30^{\circ} \), the opposite side to \( \angle I \) is \( GH \) and the adjacent side is \( HI \). So \( \tan(30^{\circ})=\frac{GH}{HI} \).

Step2: Recall the value of \( \tan(30^{\circ}) \)

We know that \( \tan(30^{\circ})=\frac{1}{\sqrt{3}} \). And \( HI = 7\sqrt{3} \). Substituting into the tangent formula:
\( \frac{1}{\sqrt{3}}=\frac{GH}{7\sqrt{3}} \)

Step3: Solve for \( GH \)

Cross - multiply to get \( GH\times\sqrt{3}=7\sqrt{3}\times1 \). Then divide both sides by \( \sqrt{3} \):
\( GH=\frac{7\sqrt{3}}{\sqrt{3}} = 7 \)

Answer:

\( 7 \)